Arctic curves of the octahedron equation

Francesco, P. Di; Soto-Garrido, Rodrigo

doi:10.1088/1751-8113/47/28/285204

Cited by 30 publications

(46 citation statements)

References 44 publications

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“…In [34,35] global properties have been discussed in a general context, including periodic weightings for the Aztec diamond. See also [25] for results for a certain family of periodic weightings of the Aztec diamond. To the best of our knowledge, only in case of the two-periodic weighting the fine asymptotic properties were studied.…”

Section: Example: Domino Tiling Of the Aztec Diamondmentioning

confidence: 99%

Correlation functions for determinantal processes defined by infinite block Toeplitz minors

Berggren

Duits

2019

Advances in Mathematics

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We study the correlation functions for determinantal point processes defined by products of infinite minors of block Toeplitz matrices. The motivation for studying such processes comes from doubly periodically weighted tilings of planar domains, such as the two-periodic Aztec diamond. Our main results are double integral formulas for the correlation kernels. In general, the integrand is a matrix-valued function built out of a factorization of the matrix-valued weight. In concrete examples the factorization can be worked out in detail and we obtain explicit integrands. In particular, we find an alternative proof for a formula for the two-periodic Aztec diamond recently derived in [20]. We strongly believe that also in other concrete cases the double integral formulas are good starting points for asymptotic studies.for a determinantal point process can be expressed in terms of determinants of matrices constructed out of a function of two variables, called the correlation kernel. For Schur processes, this correlation kernel can be written in terms of double integral formulas with explicitly known integrands. Saddle point methods are thus at our disposal for the asymptotic analysis of concrete examples. This has been a large industry in recent years and we do not attempt to provide a full list of works, but point to [1,6,32,33] and the references therein, for introductions to this subject and as general references.An important motivation for studying the extension to block Toeplitz minors comes from random tilings of planar domains or random dimer configurations with doubly-periodic weights, that was discussed in [34,35,40]. The double periodicity in the weight structure naturally leads in many cases to taking minors of block Toeplitz matrices. Being a natural and non-trivial extension of the scalar case one may therefore expect a richer structure where new phenomena can be discovered. A remarkable feature for periodically weighted dimer models is that a so-called gas region [35,40] may appear. In such a region the 2-point correlations for the height function decay exponentially with the distance. However, the integrable structure for these models is relatively unexplored and one reason for this is that the standard techniques for the scalar case are inadequate. Explicit double integrals for the kernel, even for n → ∞, are not known generally. In fact, to the best of our knowledge, such a double integral formula is only known in case of the two-periodic Aztec diamond. The first results are by Chhita and Young [13] and Chhita and Johansson [11], who found a machinery for computing the inverse Kasteleyn matrix explicitly and used that to perform an asymptotic analysis. See also [3] for further results.Of special importance to us is the recent paper [20], where one of us together with Kuijlaars took a different approach for the two-periodic Aztec diamond. Starting from the definition of non-intersecting path ensembles with general block Toeplitz transitions we showed that the correlation kernel can be related to matrix ...

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Section: Example: Domino Tiling Of the Aztec Diamondmentioning

confidence: 99%

Correlation functions for determinantal processes defined by infinite block Toeplitz minors

Berggren

Duits

2019

Advances in Mathematics

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“…This phenomenon was soon observed to be ubiquitous within the context of highly correlated statistical mechanical systems; see, for instance, [1,2,5,6,7,10,12,13,15,16,17,18,19,20,25,28,29,30,32,33,34,35,42,43,44,45,51,54,57,61]. In particular, Cohn-Kenyon-Propp developed a variational principle [12] that prescribes a law of large numbers for random domino tilings on almost arbitrary domains, which was used effectively by to explicitly determine the arctic boundaries of uniformly random lozenge tilings on polygonal domains.…”

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confidence: 99%

Arctic boundaries of the ice model on three-bundle domains

Aggarwal

2019

Invent. math.

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In this paper we consider the six-vertex model at ice point on an arbitrary threebundle domain, which is a generalization of the domain-wall ice model on the square (or, equivalently, of a uniformly random alternating sign matrix). We show that this model exhibits the arctic boundary phenomenon, whose boundary is given by a union of explicit algebraic curves. This was originally predicted by Colomo-Sportiello in 2016 as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries, called the (geometric) tangent method. Our proof uses a probabilistic analysis of non-crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case, which might be of independent interest.

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“…We just do the computations here to show how it fits into our particular framework. See also [33,11,19] for similar proofs in the case of the octahedron and cube recurrences.…”

Section: Limit Shapesmentioning

confidence: 98%

“…• In Section 4 we define taut configurations, and prove that the solution of Kashaev's recurrence is the partition function of these configurations. We compute some limit shapes of the model by a now standard technique [33,11,29,19]. We also show that in characteristic 2 the model reduces to the cube groves of [5].…”

Section: Theorem For Any Free-fermionic Cmentioning

confidence: 99%

The free-fermionic C2(1) loop model, double dimers and Kashaev's recurrence

Melotti

2018

Journal of Combinatorial Theory, Series A

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We study a two-color loop model known as the C (1) 2 loop model. We define a free-fermionic regime for this model, and show that under this assumption it can be transformed into a double dimer model. We then compute its free energy on periodic planar graphs. We also study the star-triangle relation or Yang-Baxter equations of this model, and show that after a proper parametrization they can be summed up into a single relation known as Kashaev's relation. This is enough to identify the solution of Kashaev's relation as the partition function of a C (1) 2 loop model with some boundary conditions, thus solving an open question of Kenyon and Pemantle [29] about the combinatorics of Kashaev's relation. 1 Kashaev's initial equation contained a +4 instead of a −4 coefficient for the last terms, but one can easily get from one to another, for instance by multiplying g by −1 at a vertex of the cube and its three neighbors.

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Arctic curves of the octahedron equation

Cited by 30 publications

References 44 publications

Correlation functions for determinantal processes defined by infinite block Toeplitz minors

Correlation functions for determinantal processes defined by infinite block Toeplitz minors

Arctic boundaries of the ice model on three-bundle domains

The free-fermionic C2(1) loop model, double dimers and Kashaev's recurrence

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